Question

Ages of Pennies We collected 50 pennies and recorded their ages, by calculating AGE = CURRENT YEAR - YEAR ON PENNY. \(\begin{array}{rrrrrrrrrr}5 & 1 & 9 & 1 & 2 & 20 & 0 & 25 & 0 & 17 \\ 1 & 4 & 4 & 3 & 0 & 25 & 3 & 3 & 8 & 28 \\ 5 & 21 & 19 & 9 & 0 & 5 & 0 & 2 & 1 & 0 \\\ 0 & 1 & 19 & 0 & 2 & 0 & 20 & 16 & 22 & 10 \\ 19 & 36 & 23 & 0 & 1 & 17 & 6 & 0 & 5 & 0\end{array}\) a. Before drawing any graphs, try to visualize what the distribution of penny ages will look like. Will it be mound-shaped, symmetric, skewed right, or skewed left? b. Draw a relative frequency histogram to describe the distribution of penny ages. How would you describe the shape of the distribution?

Short answer

Answer: The shape of the distribution based on the relative frequency histogram of penny ages is skewed right.

Step-by-step solution

Organize the data

To understand the distribution better, we can start by organizing the data in increasing order: 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 12, 14, 14, 14, 16, 17, 17, 19, 20, 21, 23, 24, 25, 28, 41.

Visualize the distribution before drawing any graphs

Based on the organized data, we can see that there is a higher frequency of lower ages (closer to 0) and a lower frequency of higher ages. We can expect that the distribution will be skewed right since there is a long tail on the right side and most of the data points are on the left side.

Divide the dataset into different intervals

To draw a relative frequency histogram, we need to divide the dataset into different intervals (also called bins) to have a better representation of frequencies. Since the maximum age is 41, we can create 9 intervals with a step of 5: 0-4, 5-9, 10-14, 15-19, 20-24, 25-29, 30-34, 35-39, 40-44.

Calculate the relative frequency for each interval

Now, we need to determine how many data points fall into each interval, divide it by the total number of data points (50), and then multiply by 100 to get the relative frequency (in percentage) for each interval: 1. 0-4 : 20/50 * 100 = 40% 2. 5-9 : 14/50 * 100 = 28% 3. 10-14 : 7/50 * 100 = 14% 4. 15-19 : 3/50 * 100 = 6% 5. 20-24 : 4/50 * 100 = 8% 6. 25-29 : 2/50 * 100 = 4% 7. 30-34 : 0/50 * 100 = 0% 8. 35-39 : 0/50 * 100 = 0% 9. 40-44 : 1/50 * 100 = 2%

Draw a relative frequency histogram

On the horizontal axis, put the intervals, and on the vertical axis, put the relative frequency (in percentage). For each interval, draw a bar whose height corresponds to the relative frequency.

Describe the shape of the distribution

Based on the relative frequency histogram, we can conclude that the distribution of penny ages is skewed right, as we expected in Step 2. This means most of the pennies have lower ages, with few older pennies in the dataset.

Key concepts

Frequency Histogram

Visualizing data effectively is crucial for understanding the underlying patterns and trends, and a frequency histogram is one of the most fundamental tools in this endeavor. A frequency histogram is a type of bar graph that represents the frequency of data points within specified ranges, called bins. When dealing with a large set of numbers, like ages of pennies, organizing them into a histogram allows for a clearer picture of how the pennies are distributed across different age intervals.

Each bar in the histogram corresponds to a bin and the height of the bar signifies the number of observations falling within that bin's range. By providing a simple graphical representation, histograms can quickly communicate the data's structure. For instance, a histogram of penny ages can show at a glance whether there are more new or old pennies in a given collection. Easy to understand and interpret, frequency histograms are a staple in statistical visualization for their ability to condense data into a coherent form.

Skewed Distribution

Understanding the symmetry of a data set is crucial, and the concept of skewed distribution provides valuable insights into the asymmetry of data. When data points cluster more to one side of the scale, the distribution is said to be 'skewed'. There are two types of skewness - a distribution is 'skewed right' if there's a tail on the right side, indicating a cluster of lower values, and fewer high-value outliers. Conversely, a 'skewed left' distribution has a tail towards the left side.

Referring to our penny scenario, the prediction before drawing the histogram was that the distribution would be skewed right. This insight comes from noticing the higher frequency of younger pennies, and fewer pennies that are old - mirroring the tail of the distribution extending to the right. Skewed distributions are common in real-world scenarios and recognizing them helps us understand the nature of data, influencing decisions and predictions. For instance, skewed data could affect the strategy for collecting coins if one were to aim for a particular age range.

Data Visualization in Statistics

Data visualization is a powerful aspect of statistical analysis, enabling individuals to see patterns, outliers, and trends that are not readily apparent in raw data. By transforming numbers into graphical representations, visualizations make complex data more accessible and understandable. Tools like bar charts, line graphs, scatter plots, and frequency histograms are part of this visualization toolkit.

In the context of penny ages, visualizing data reveals not just the average age, but also the distribution and variability within the pennies collected. Effective data visualization facilitates better comprehension, quicker analysis, and can guide further statistical inquiry or hypothesis testing. It can also be a persuasive tool, providing a graphical argument for a particular interpretation of the data. As such, knowledge of data visualization is essential for students and professionals alike, fostering a deeper engagement with the data they encounter.

Relative Frequency

While the frequency histogram tells us how often something happens, relative frequency gives context to those numbers by considering the size of the dataset. It's represented as a percentage that shows how large each bin is relative to the total number of observations. This is particularly useful when comparing datasets of different sizes or to give a standardized view of the data.

To calculate the relative frequency, as shown in Step 4 of the exercise, you divide the number of data points in each bin by the total number of data points and then multiply by 100. In the pennies example, computing the relative frequencies helps us understand not just how many pennies are in each age group, but how that group compares to the entire collection. This added layer of information can provide insight into the probability distribution of age, an essential aspect in understanding the overall characteristics of the dataset.